Calc Final

Front 1

Limit def in symbols

Back 1

lim f(x)= L for every E>0, there is
x->a a delta>0 so that
0<|x-a| <delta then |f(x)-L| <E

Front 2

Verbal def of the limit

Back 2

We say that Lim x->a f(x)=L if we can make the values of f(x) arbitrarily close to the number L by taking x sufficiently close to the number a (excluding the point at x=a)

Front 3

Methods for computing limit

Back 3

simplify
factor and cancel
multiply by conjugate
divide by x x->infinity
l'hospital's rule 0/0 infinity/infinity

Front 4

horizontal asymptote

Back 4

y=b for f(x) if one of the following is true: lim x-> +- inf. f(x)=b

Front 5

vertical asymptote

Back 5

x=a for f(x) if one of the following limits is infinite: lim x-> +- a f(x)

Front 6

indeterminates

Back 6

0/0 +- inf/+-inf 0(inf) 1^inf inf-inf

Front 7

L'Hospital

Back 7

0/0 inf/inf 0(inf) 1^inf

f(x)g(x)= f(x)/1/g(x)
f(x)^g(x)= e^g(x)ln(f(x))

Front 8

Continuity def

and 3 things

Back 8

lim x->a f(x)=f(a)

1. f(a) exists
2. the limit exists
3. 1 and 2 are the same number

Front 9

IVT

Back 9

if f is continuous on [a,b] and w is a number between f(a) and f(b), there isa t least one c in [a,b] so that f(c)= w

-if f is continuous, the range of a closed interval is an interval
-if f is continuous, and f(x1)>0, f(x2)<0, then there is a c between x1 and x2 where f(c)=0...at least one root

Front 10

continuous vs differentiable

Back 10

if f is differentiable at x=a, it is continuous at x=a. If f is continuous at x=a, we don't know if it is differentiable at x=a. That is, "all differentiable function are continuous, but not all continuous functions are differentiable."

Front 11

Derivative def

Back 11

f'(x)= lim h->0 f(x+h)-f(x)/h

Front 12

inverses

Back 12

if the point (a,b) is on the graph of f, then 1) the point (b,a) is on the graph of f^-1, and 2) f'(b)=1/f'(a)

Front 13

Rolle's theorem

Back 13

If f is continuous on [a,b] and f is differentiable on (a,b) and f(a)=f(b) then there is a c between a and b for which f'(c)=0

Front 14

connection between root of a function and its derivative

Back 14

-if f'(x) does not equal 0 on (a,b), then f(x) can have at most 1 root on [a,b]
-if f'(x)=0 has one solution on (a,b), the f can have at most 2 roots on [a,b]
-if f'(x)=0 has two solutions on (a,b) then f can have at most 3 roots on [a,b}

Front 15

MVT def

Back 15

if f is continuous on [a,b] and differentiable on (a,b) then there is a c in the interval (a,b) so that
f'(c)=f(b)-f(a)/b-a

Front 16

The racetrack principle

Back 16

if f(a)=g(a) and f'(x)>g'(x) for all x in [a,b] then f(x) >g(x) for x in [a,b]

Front 17

linear approximation

Back 17

the linearization of f at x=a is the tangent line approximation to f at x=a: y-f(a)=f'(a)(x-a)

Front 18

differentials
delta y dy

Back 18

if x changes from x to delta x, then there is a corresponding change in f:
delta y=f(x-delta x)-f(x)
dy=f'(x)dx

Front 19

EVT

Back 19

if f is continuous on [a,b] then f will attain a global max and global min on [a,b]. These points will be either at critical points or at endpoints

Front 20

Local:
let x=a be a critical point of f
first and second derivative test

Back 20

first derivative test:
-if f'(x) changes sign at x=a then we have a local min (- to +), or a local max (+ to -)
-if f'(x) does not change sign at x=a...

second derivative test:
-if f''(a)>0, we have a local min at x=a
-if f''(a) <0, we have a local max at x=a
-if f''(a)=0 the test is inconclusive

Front 21

Global:

Back 21

-on a closed interval: use EVT, build a chart using endpoints and critical points
-not on a close interval: if f'(x) only changes sign once at x=a, then x=a, y=f(a) is either a global min (-to+) or a global max (+to-)

Front 22

Newton's method

Back 22

f(x)=0 for x
xi+1=xi-f(xi)/f'(xi)

Front 23

cf

Back 23

cf'

Front 24

f+-g

Back 24

f'+-g'

Front 25

fg

Back 25

f'g+fg'

Front 26

f(g(x))

Back 26

f'(g(x))g'(x)

Front 27

f/g

Back 27

f'g-g'f/g^2

Front 28

f(x)^g(x)

Back 28

log diff

Front 29

c

Back 29

0

Front 30

x^n

Back 30

nX^n-1

Front 31

e^x

Back 31

e^x

Front 32

a^x

Back 32

a^x(ln(a))

Front 33

ln|x|

Back 33

1/x

Front 34

loga(x)

Back 34

1/x (1/ln(a))

Front 35

sin(x)

Back 35

cos(x)

Front 36

cos(x)

Back 36

-sin(x)

Front 37

tan(x)

Back 37

sec2(x)

Front 38

sec(x)

Back 38

sec(x)tan(x)

Front 39

csc(x)

Back 39

-csc(x)cot(x)

Front 40

cot(x)

Back 40

-csc2(x)

Front 41

sin-1(x)

Back 41

1/sqr(1-x2)

Front 42

cos-1(x)

Back 42

-1/sqr(1-x2)

Front 43

tan-1(x)

Back 43

1/1+x2

Front 44

cf

Back 44

cF

Front 45

f+-g

Back 45

F+-G

Front 46

c

Back 46

cx

Front 47

x^n

Back 47

1/n+1 (x^n+1)

Front 48

1/x

Back 48

ln|x|

Front 49

e^x

Back 49

e^x

Front 50

cos(x)

Back 50

sin(x)

Front 51

sin(x)

Back 51

-cos(x)

Front 52

sec2(x)

Back 52

tan(x)

Front 53

sec(x)tan(x)

Back 53

sec(x)

Front 54

1/sqr(1-x2)

Back 54

sin-1(x)

Front 55

1/1+x2

Back 55

tan-1(x)